Nonparametric mixed exponentially weighted moving average-moving average control chart

This research designed a distribution-free mixed exponentially weighted moving average-moving average (EWMA-MA) control chart based on signed-rank statistic to effectively identify changes in the process location. The EWMA-MA charting statistic assigns more weight to information obtained from the recent \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w$$\end{document}w samples and exponentially decreasing weights to information accumulated from all other past samples. The run-length profile of the proposed chart is obtained by employing Monte Carlo simulation techniques. The effectiveness of the proposed chart is evaluated under symmetrical distributions using a variety of individual and overall performance measures. The analysis of the run-length profile indicates that the proposed chart performs better than the existing control charts discussed in the literature. Additionally, an application from a gas turbine is provided to demonstrate how the proposed chart can be used in practice.

detecting shifts in the location parameter.The signed-rank statistic used in this study is found to be more efficient and has greater statistical power compared to sign statistic because it considers the observations' magnitude in addition to the signs (see, Graham et al. 40 and Hollander et al. 41 ).The proposed control chart would be an alternative choice for practitioners and applicable in situations where the process distribution is either unknown or non-normal when quick detection of shift in the process location is of paramount interest.For instance, the proposed methodology can be used to monitor the inside diameters of piston rings manufactured by a forging process considered by Graham et al. 42 , the flow width of resist in hard-bake process used by Alevizakos et al. 43 , the filled liquid volume of soft drink beverage bottles considered by Raza et al. 44,45 .Moreover, to signify the practical implementation of the proposal a gas turbine data is used to monitor the ambient humidity which is an important characteristic that effects the CO and NOx emissions.
The rest of the paper is organized as follows: Sect."The design structure of the EWMA-MA signed-rank control chart" presents the charting structure of the distribution-free mixed EWMA-MA signed-rank control chart.Section "Performance evaluation" assesses the run-length performance of the proposed chart under various symmetric distributions.In Sect."Comparative study", a comparative study is carried out to evaluate the performance of the proposal compared to its competitors.To validate the proposed chart's practicability, an application to monitor the ambient humidity generated from the gas turbine is presented in Sect."Real-life example".Finally, Sect."Summary and conclusion" concludes the paper.

The design structure of the EWMA-MA signed-rank control chart
Consider a quality characteristic (X) with known median (θ) as a target value.Let X ij be the i th observation within the j th sample or subgroup of size n(> 1) , where i = 1, 2, . . ., n and j = 1, 2, . . . .Furthermore, R + ij is the rank assigned to the absolute differences from the targeted value θ , i.e.X ij − θ .The signed-rank statistic SR j is defined as: where The SR statistic is a linear function of the Mann-Whitney statistic M + n , i.e., SR = 2M + n − n(n + 1)/2 (for more details, see Gibbons and Chakraborti 46 ).The SR statistic has a zero mean and n(n + 1)(2n + 1)/6 vari- ance.The distribution-free mixed EWMA-MA signed-rank statistic is developed by integrating MA statistic into EWMA statistic.The moving average statistic MA j of span w at time j is: The mean of the moving average is E MA j = µ 0 = 0 .The monitoring statistic of EWMA-MA signed-rank control chart is defined as: where is the smooting constant (0 < < 1) .The initial value of E SR j is taken as the mean of SR statistic, i.e.E SR 0 = E(SR) = µ 0 = 0. Now, the statistic E SR j can also be expanded as: The in-control (IC) expected value of the E SR j is: To obtain the variance of the statistic E SR j , we apply variance on both sides of Eq. ( 4) and get: (1) The center line (CL) , lower control limit (LCL) , and upper control limit (UCL) for the EWMA-MA signed- rank control chart are determined as: where L > 0 is the width of the control limits.The monitoring statistics E SR j are plotted against their respective control limits.If either E SR j ≥ UCL or E SR j ≤ LCL , then process is considered as out-of-control (OOC).In such a case, it is crucial for a quality practitioner to thoroughly investigate the process and detect the assignable cause(s).On the other hand, if LCL< E SR j < UCL , the process is declared as stable or in-control (IC), indicating that no shift has been detected and the process is operating within acceptable limits.The suggested EWMA-MA control chart encompasses the nonparametric EWMA control chart introduced by Amin and Searcy 22 when w = 1 and the MA signed-rank control chart for = 1 .Knoth et al. 47 criticized mixed control charts by claiming that these control charts assign more weights to past data values than current ones.Recently, contrary to the findings of Knoth et al. 47 , Alevizakos et al. 48evaluated the performance of the various mixed memory type EWMA control charts and showed that these charts have superior OOC zero-state and steady-state run length performance, especially for smaller to moderate shifts.It is to be noted that the EWMA-MA statistic assigns more weight to the current ' w ' observations while exponentially decreasing weights to the rest of the observations.It is due to the reliance of the MA statistic on the current w observations.As a result, the weighting structure EWMA-MA statis- tic matches with the conventional EWMA for observations older than w , i.e. their weight decreases exponentially.

Performance evaluation
To evaluate the effectiveness of the control chart, the average run-length (ARL) is commonly used to quantify the average number of samples displayed on a control chart before the occurrence of the first OOC signal 49 .The IC and OOC average run-length are denoted by ARL 0 and ARL 1 , respectively.If the process is IC, ARL 0 is typically set to be sufficiently large to minimize the false alarm.Conversely, the ARL 1 should be small to quickly identify any process shift.To gain deeper insights into the run-length distribution and evaluate the performance of the chart, additional performance metrics such as the standard deviation of run-length ( SDRL) and median run-length ( MRL) are used in the literature 50-53 .The performance metrics discussed earlier are used for specific process shifts.To assess overall performance for a range of shifts, additional metrics like the average extra quadratic loss (AEQL) and relative mean index (RMI) are computed in this study.The AEQL is the weighted average of ARL calculated for different shifts considered in a process.More information about AEQL may be found in Raza et al. 39 and Malela-Majika 54 .The algebraic expression of AEQL is as follows: where, δ represents the shift's magnitude, ARL(δ) is a ARL value at a specific shift δ in a process, δ max and δ min indicate the highest and lowest values of the shifts taken into consideration, respectively.A smaller AEQL value indicates its ability to identify process shifts quickly.Han and Tsung 55 introduced the RMI which is based on the relative difference of the ARL values.RMI is mathematically defined as: ( 6) where ARL (δ i ) refers to the ARL value of the control chart under the specified shift, and ARL * (δ i ) denotes the smallest ARL value across all the control charts that are considered for the comparison under the shift δ i .N represents the total number of shifts considered for comparative purposes.The superiority of the control chart is determined by its lower RMI value when compared to other control charts.
In this research, a Monte Carlo simulation is used as a computational technique to obtain numerical findings for evaluating the performance of the control charts.With the help of R software, 10,000 iterations are used to determine the ARL , SDRL , and MRL values.To achieve the intended ARL 0 , several combinations of the design parameters ( , w) and the limit coefficient (L) are tested during the simulation method.The charting statistics SR j is of a discrete nature, so it is not always possible to achieve the exact desired, ARL 0 .Therefore, we endure the 1% of variation in desired ARL 0 .The run-length characteristics of the EWMA-MA signed-rank control chart are calculated using the following algorithm: Calculating the IC run-length profile i. Choose a specific distribution, such as the normal distribution with mean µ 0 and variance σ 2 to produce 10,000 random samples of size n.ii.Select suitable values for and w. iii.To achieve a desired ARL 0 , such as 370, we must identify the appropriate L value while maintaining n , , and w as constants.iv.Calculate the SR j statistic from Eq. ( 3) and subsequently compute the monitoring statistic E SR j .v. Compare the monitoring statistic E SR j with the respective control limits given in Eq. ( 9).vi.The number of samples is recorded before the monitoring statistic first exceeds the control limit, which is defined as a run-length.vii.Steps 1 through 6 are repeated 10,000 times to acquire ARL.viii.If the value of ARL is approximately equal to the desired ARL 0 , proceed to compute SDRL and MRL, then move on to the next steps.Otherwise, change the value of L and repeat Steps 1 to 7 until the desired ARL 0 is achieved.

Calculating the OOC run-length profile
ix.A process shift (δ = 0) is introduced to obtain a test sample of size n to simulate the OOC process state, i.e. generating samples from a normal distribution with a shifted mean µ 1 = µ 0 + δσ and variance σ 2 . (11 Table 1.The limit coefficient (L) values for various combinations of (n, w, ) at ARL 0 ∼ = 370.x.To determine the run-length characteristics under the OOC scenario, Steps 4 through 7 are iteratively executed 10,000 times and subsequently the values of ARL 1 , SDRL 1 , and MRL 1 are obtained based on the OOC run-lengths.xi.After computing the value of ARL 1 for all shifts examined in the study, the AEQL is calculated as a measure of the overall performance evaluation for the EWMA-MA signed-rank control chart.
The values of the limit coefficient (L) for the EWMA-MA signed-rank control chart were obtained by using the aforementioned algorithm for various combinations of sample size (n) , span (w) , and smoothing parameter ( ) under the fixed ARL 0 ∼ = 370 .The results under various parameter settings are displayed in Table 1 which are summarized as: i.For a specified value of n and , the value of the limit coefficient L decreases as w increases to achieve the desired ARL 0 .For example, if we fix n = 10 and = 0.05 , then the value of L is 2.304 for w = 5 and it decreases to 2.205 for w = 10.ii.Similarly, if n and w are fixed, the value of the limit coefficient increases with .For instance, with n = 12 and w = 5 , the values of L are 2.305 and 2.481 for = 0.05 and 0.10 , respectively.iii.The value of the limit coefficient changes slightly with sample size n by keeping other design parameters as fixed.
The performance and robustness of the nonparametric EWMA-MA signed-rank control chart were determined by assessing shift detection ability for a range of symmetrical distributions, including the standard normal distribution N(0, 1) ; the Logistic distribution, LG 0, √ π ; the Student's t distribution, t(4) and t(10) ; the Laplace distribution, Laplace 0, 1 √ 2 ; as well as the contaminated normal (CN) .The CN is defined as the combination of two normal distributions with common mean µ and different variances, i.e., (1 − β)N µ, σ 2 1 + βN µ, σ 2 2 , where σ 1 = 2σ 2 and proportion of contamination is β = 0.10.For ARL 0 ∼ = 370 , n = 10 , and various combina- tions of design parameters ( , w, L ), Tables 2, 3, 4, 5, 6 and 7 display the computed run-length characteristics of the proposal under these distributions.The following observations are made from Tables 2, 3, 4, 5, 6 and 7: i.The results depicts that the IC run-length distribution of the EWMA-MA signed-rank chart remains the same across the various process distributions considered in this study, which is in line with the distribution-free control charting theory.ii.The OOC run length performance of the proposed chart to detect smaller shifts improves as the value of w increases under a fixed sample size n and sensitivity parameter .For instance, for n = 10 , = 0.05 and specified shift size δ = 0.05 , the ARL 1 value of the proposed chart decreases to 139.1 from 143.7 and MRL 1 decreases to 98 from 106 when w increases from 5 to 10 under student's t distribution with 10 degrees of  2 and 5).In general, the choice of w depends on the shift size that needs to be detected quickly.If smaller shift is of interest then a large value of w should be taken and conversely, a lower value is beneficial for larger shifts.iii.The OOC run-lengths tend to increase with for small to moderate shifts (δ ≤ 1.0) under fixed n and w .
For example, under the shifted process with δ = 0.10 , n = 10 , and w = 5 , the ARL 1 increases to 57.1 from 46.4 and MRL 1 increases to 43 from 37 when increases from 0.05 to 0.10 under the CN distribution (cf.Tables 2 and 3).

Comparative study
The performance of the EWMA-MA signed-rank control chart is evaluated and compared with other competing control charts like MA sign (MA-SN) and MA signed rank (MA-SR) by Pawar et al. 56 , EWMA sign (EWMA-SN) by Yang et al. 26 , EWMA signed-rank (EWMA-SR) by Graham et al. 42 , and mixed EWMA-CUSUM sign   57 .The comparison of the OOC run-length distribution is made under various symmetrical distributions based on different performance metrics such as ARL 1 ,SDRL 1 , and MRL 1 for a range of shifts (δ) in the process.Moreover, the AEQL and RMI are used to assess the overall effectiveness of the proposed control chart in comparison to its competitors.For a rational comparison between the EWMA-MA signed-rank and existing control charts, the IC run-length is fixed at ARL 0 = 370 with a sample size n = 10 .The MA-SN and MA-SR control charts were constructed by setting w = 5 , with k = 3.10 and 2.849 , respectively.Likewise, = 0.05 with k = 2.675 and 2.481 were used to set up the EWMA-SN and EWMA-SR control charts, respectively.The MEC-SN control chart was computed using the design parameters = 0.05,k = 0.5 , and h = 51.28 .Furthermore, the EWMA-MA signed rank control chart was calculated using the parameter settings w = 5 , = 0.05 , and L = 2.304 .The ARL and SDRL values of each control chart are given in the first row of Table 8, while MRL is provided in the second row.The minimum values of ARL 1 ,AEQL , and RMI are indicated by bold fonts.The following observations are made from Table 8: i.As the magnitude of the shift increases, the run-length properties associated with OOC conditions exhibit a rapid decrease.ii.The EWMA-MA signed-rank control chart outperforms its counterparts in detecting a specific shift in the process mean, regardless of distribution type.iii.The proposed chart exhibits superior overall effectiveness in detecting a range of shifts with smaller values of AEQL and RMI as compared to the existing control charts.

Real-life example
To demonstrate the applicability and relevance of the EWMA-SR singed-rank chart to real-life scenarios, an industrial dataset of a gas-turbine located in Türkiye 58 was taken.The dataset consists of 36733 observations covering the period 2011 to 2016 from 11 sensors at hourly intervals.The dataset includes the following main parameters: ambient temperature (AT), ambient humidity (AH), ambient pressure (AP), gas turbine exhaust pressure, air filter differential pressure, turbine inlet temperature, turbine after temperature, turbine energy yield (TEY), carbon monoxide (CO) emissions, compressor discharge pressure, and nitrogen oxide (NOx) emissions.Many researchers used different key factors of combined cycle power plants in their studies to monitor the energy output of the plant.For example, Nawaz and Han 59 examined the AP as a variable of interest and its impact on the overall performance of the power plant.Similarly, Raza et al. 39 utilized the AT as a variable of interest to demonstrate how it affects the overall performance of a power plant.In this study, ambient humidity (AH) is selected as a variable of interest that can significantly affect the performance of gas-turbine, i.e.The higher AH in combustion air lowers NOx emissions by reducing peak flame temperature and enhances combustion efficiency, resulting in lower CO emissions in gas turbines.The sustained higher AH level for keeping the emissions in gas turbines at a lower level can contribute to environmental goals by lowering harmful pollutants like NOx and CO.The average and standard deviation of AH are 0.72 and 0.15 , respectively.The coefficient of skewness is www.nature.com/scientificreports/Table 8.The run-length characteristics (the first row contain ARL 1 s with SDRL 1 s in parenthesis, while MRL 1 s are in second row) of the existing MA-SN, EWMA-SN, MA-SR, EWMA-SR, MEC-SN, and proposed EWMA-MA(SR) control charts for n = 10 at ARL 0 ≈ 370.Significant values are in bold.

Summary and conclusion
In circumstances where the underlying distribution of a quality characteristic being monitored is unknown, nonparametric control charts offer a reliable and highly effective mechanism for monitoring a process.This study presented the distribution-free mixed EWMA-MA control chart, which is based on the signed-rank statistic for efficient detection of shifts in the process location.The run-length profile of the proposal is studied and compared with several competing control charts using extensive Monte Carlo simulations under a variety of symmetrical  process distributions.Based on the obtained results, it is found that the proposed chart is more effective not only for detecting a specified shift in the process location but also in its overall ability to detect a range of shifts.In addition, a real-life example is provided to further validate the proposed chart's practicability and effectiveness in identifying process shifts in comparison to other competing control charts.The effectiveness of the proposed charting structure can be further explored for monitoring the process dispersion and joint monitoring of location and dispersion parameters.Moreover, a comprehensive investigation can be carried out to find the optimal values of the smoothing parameter and span for various shifts of interest.

Figure 2 .Figure 3 .
Figure 2. Nonparametric MA sign control chart of AH data.

Figure 4 .
Figure 4. Nonparametric mixed EWMA-CUSUM sign control chart of AH data.

Figure 5 .
Figure 5. Nonparametric MA signed-rank control chart of AH data.

Figure 6 .
Figure 6.Nonparametric EWMA signed-rank control chart of AH data.

Figure 7 .
Figure 7. Distribution-free mixed EWMA-MA signed-rank control chart of AH data.